Mathematical Induction 1 Mathematical Induction Example Show that

Mathematical Induction 1

Mathematical Induction: Example Ø Show that any postage of ≥ 8¢ can be obtained using 3¢ and 5¢ stamps. Ø First check for a few particular values: 8¢ = 3¢ + 5¢ 9¢ = 3¢ + 3¢ 10¢ = 5¢ + 5¢ 11¢ = 5¢ + 3¢ 12¢ = 3¢ + 3¢ Ø How to generalize this? 2

Mathematical Induction: Example • Let P(n) be the sentence “n cents postage can be obtained using 3¢ and 5¢ stamps”. • Want to show that “P(k) is true” implies “P(k+1) is true” for any k ≥ 8¢. • 2 cases: 1) P(k) is true and the k cents contain at least one 5¢. 2) P(k) is true and the k cents do not contain any 5¢. 3

Mathematical Induction: Example • Case 1: k cents contain at least one 5¢ coin. Replace 5¢ coin by two 3¢ coins k cents 5¢ k+1 cents 3¢ 3¢ • Case 2: k cents do not contain any 5¢ coin. Then there at least three 3¢ coins. k cents 3¢ 3¢ 3¢ Replace three 3¢ coins by two 5¢ coins k+1 cents 5¢ 5¢ 4

Domino Effect • Mathematical induction works like domino effect: • Let P(n) be “The nth domino falls backward”. • If (a) “P(1) is true”; (b) “P(k) is true” implies “P(k+1) is true” Then P(n) is true for every n 5

Principle of Mathematical Induction Let P(n) be a predicate defined for integers n. Suppose the following statements are true: 1. Basis step: P(a) is true for some fixed a Z. 2. Inductive step: For all integers k ≥ a, if P(k) is true then P(k+1) is true. Then for all integers n ≥ a, P(n) is true. 6

Example: Sum of Odd Integers Ø Proposition: 1 + 3 + … + (2 n-1) = n 2 for all integers n≥ 1. Ø Proof (by induction): 1) Basis step: The statement is true for n=1: 1=12. 2) Inductive step: Assume the statement is true for some k≥ 1 (inductive hypothesis) , show that it is true for k+1. 7

Example: Sum of Odd Integers Ø Proof (cont. ): The statement is true for k: 1+3+…+(2 k-1) = k 2 (1) We need to show it for k+1: 1+3+…+(2(k+1)-1) = (k+1)2 (2) Showing (2): 1+3+…+(2(k+1)-1) = 1+3+…+(2 k+1) = by (1) 1+3+…+(2 k-1)+(2 k+1) = k 2+(2 k+1) = (k+1)2. We proved the basis and inductive steps, so we conclude that the given statement true. ■

Important theorems proved by mathematical induction Ø Theorem 1 (Sum of the first n integers): For all integers n≥ 1, Ø Theorem 2 (Sum of a geometric sequence): For any real number r except 1, and any integer n≥ 0, 9

Example (of sum of the first n integers) Ø In a round-robin tournament each of the n teams plays every other team exactly once. What is the total number of games played? Solution on the board. 10

Proving a divisibility property by mathematical induction • Proposition: For any integer n≥ 1, 7 n - 2 n is divisible by 5. (P(n)) • Proof (by induction): 1) Basis step: The statement is true for n=1: (P(1)) 71 – 21 = 7 - 2 = 5 is divisible by 5. 2) Inductive step: Assume the statement is true for some k≥ 1 (P(k)) (inductive hypothesis) ; show that it is true for k+1. (P(k+1)) 11

Proving a divisibility property by mathematical induction Ø Proof (cont. ): We are given that P(k): 7 k - 2 k is divisible by 5. (1) Then 7 k - 2 k = 5 a for some a Z. (by definition) (2) We need to show: P(k+1): 7 k+1 - 2 k+1 is divisible by 5. (3) 7 k+1 - 2 k+1 = 7· 7 k - 2· 2 k = 5· 7 k + 2·(7 k - 2 k) = 5· 7 k + 2· 5 a (by (2)) = 5·(7 k + 2 a) which is divisible by 5. (by def. ) Thus, P(n) is true by induction. ■ 12